Processing your ADCP data using structure function techniques

To calculate the dissipation rate at a specific range bin and a specific time ensemble:

1. Extract or compute the along-beam bin center separation [$r_0$] based on the instrument geometry
2. Calculate the along-beam velocity fluctuation time-series in each bin, [Failed to parse (syntax error): {\displaystyle v’(n, t)} ] from the along-beam velocity data that has met the QC criteria (i.e. the data in Level 2 of the netcdf file)
3. Select the maximum distance ($r_{max}$) over which to compute the structure function based on conditions of the flow (e.g., expected max overturn). The corresponding number of bins are [$n_{max}=r_{max}/\delta r_0$], where $\delta r_0$ is the radial separation between bins.
4. Calculate the structure function $D$ for all possible bin separations $\delta$ using either a bin-centred difference scheme or a forward-difference scheme. Some things to consider are: [SHOULD WE NEED TO INCLUDE THESE HERE?]
   • Including $D(n,\delta )$ for $\delta =1$ may be inappropriate since the velocity estimates from adjacent bins are not wholly independent, therefore the impact of its inclusion should be evaluated
   • Keep a record of the number of instances when the squared velocity difference is evaluated for each bin $n$ and separation distance $\delta r_0$ and their distribution because they are potential quality control metrics
   • The impact of additional quality criteria can also be tested e.g. valid data requirements for all intermediate separation distances, so for a forward-difference scheme with $n=2$ and $\delta =5$, require all data in bins 2 to 7 to meet Level 1 QC requirements for the profile to be included when averaging to calculate $D(n,\delta )$
5. Perform a regression of $D(n,\delta )$ against $(\delta r_0{)}^{2/3}$ for the appropriate range of bins and $\delta$r₀ separation distances. [THE FOLLOWING ITEMS ARE CONFUSING. SINCE THIS IS BEST PRACTICE, CAN WE JUST RECOMMEND ONE METHOD?]
   1. If $D(n,\delta )$ was evaluated using a forward-difference scheme, the regression is done for the combined data from all bins in the selected range, hence the maximum number of $D(n,\delta )$ values for each separation distance will be the number of bins in the range less 1 for $\delta$ = 1, reducing by 1 for each increment in $\delta$, with the regression ultimately yielding a single $\epsilon$ value for the data segment
   2. If $D(n,\delta )$ was evaluated using a bin-centred difference scheme, the regression can either be done:
      • for each bin individually, with a single $D(n,\delta )$ for each separation distance, ultimately yielding an $\epsilon$ for each bin; or
      • by combining the data for all of the bins, with each separation distance having a $D(n,\delta )$ value for each bin, with the regression again ultimately yielding a single $\epsilon$ value for the data segment
   3. The regression is typically done as a least-squares fit, either as:
      
      $D=a_0+a_1(\delta r_0{)}^{2/3}$; or as
      $D=a_0+a_1(\delta r_0{)}^{2/3}+a_3((\delta r_0{)}^{2/3}{)}^3$
      
      the former being the canonical method that excludes non-turbulent velocity differences between bins, whereas the latter is a modified method that includes non-turbulent velocity differences between bins due to any oscillatory signal (e.g. surface waves, motion of the ADCP on a mooring).
6. Use the coefficient $a_1$ to calculate $\epsilon$ as
   
   $\epsilon ={\left(\frac{a_1}{C_2}\right)}^{2/3}$
   
   where $C_2$ is an empirical constant, typically taken as 2.0 or 2.1 [LINK TO A CONCEPTS OR FUNDAMENTALS PAGE ABOUT THIS].
7. Use the coefficient $a_1$ (the intercept of the regression) to estimate the noise of the velocity observations and compare to the expected value based on the instrument settings. [MOVE TO QA2 STEPS?]

PERHAPS WE CAN INCLUDE A FIGURE LIKE THIS TO HELP DEFINE VARIABLES.

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